I found Copenhagen perfectly safe at night, except for the usual drunkards.
The main station area is less seedy than in other European cities, and very
close to the centre. There are a lot of fast foods (either American and
Nordic style), pubs and restaurants. I found the prices average for
Continental Europe. Shopping is obviously expensive for nice porcelains and
housewares, but being a tourist city you can find a wide choice of shops.
Going around is easy with public transport; there are some tourist
double-decked buses for sightseeing and hop on-off option for one of two
days. If you have the time, visit the Christian IV observatory tower, with
a great view all over the city.
The Louisiana museum in the suburbs is devoted to Modern art; the Helsingor
castle is also not too much far north.
Enjoy your trip